Sequences and Series - Arithmetic and Geometry

If the 3rd , 5th and 8th terms of an arithmetic progression with a common difference of 3 are
three consecutive terms of a geometric progression , then what is common ratio? Help me with step-by-step.

Regards!

Answer

$$\frac{a+2d}{a}=\frac{a+5d}{a+2d}\implies a^2+4ad+4d^2=a^2+5ad\implies4d^2=ad$$
Thus, $a=4d$ or $d=0$. Since we are given $d=3$, we must have $a=12$. Thus, the common ratio is
$$\frac{a+2d}{a}=\frac{a+5d}{a+2d}=\frac32$$